Product of operators and numerical range preserving maps

Chi-Kwong Li ; Nung-Sing Sze

Studia Mathematica, Tome 173 (2006), p. 169-182 / Harvested from The Polish Digital Mathematics Library

Résumé

Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A₁,...,A_k ∈ V$ by $A₁* ⋯ * A_k = A_{i₁} ⋯ A_{iₘ}$. This includes the usual product $A₁* ⋯ * A_k = A₁ ⋯ A_k$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying $μ^{m} = 1$ such that ϕ: V → V has the form A ↦ μU*AU or $A ↦ μU*A^{t}U$, then ϕ is surjective and satisfies $W(A₁ * ⋯ *A_k) = W(ϕ(A₁)* ⋯ *ϕ(A_k))$ for all $A₁, ..., A_k ∈ V$. It is shown that the converse is true under the assumption that one of the terms in (i₁, ..., iₘ) is different from all other terms. In the finite-dimensional case, the converse can be proved without the surjectivity assumption on ϕ. An example is given to show that the assumption on (i₁, ..., iₘ) is necessary.

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Publié le : 2006-01-01

Zbl 1098.47007

EUDML-ID : urn:eudml:doc:285089